- The average rate of change of a function \(f(x)\) over an interval \([a, b]\) is calculated using the formula \(\frac{f(b) - f(a)}{b - a}\).
- For the interval \([2, 9]\), we set \(a = 2\) and \(b = 9\).
- Substituting these values into the formula yields \(\frac{f(9) - f(2)}{9 - 2}\).
- Matching this with the provided choices, the correct expression is \(\boxed{b}\).
Explanation
- Concept of Average Rate of Change
The average rate of change of a function \(f(x)\) over a closed interval \([a, b]\) measures the net change in the function's value divided by the change in the input variable over that interval. Geometrically, this represents the slope of the secant line passing through the points \((a, f(a))\) and \((b, f(b))\) on the graph of the function.
- Applying the Formula to the Given Interval
Given the interval \([a, b] = [2, 9]\), we identify \(a = 2\) and \(b = 9\). Applying the standard definition of the average rate of change:
\[ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} \]
Substituting the values \(a=2\) and \(b=9\) into the formula:
\[ \frac{f(9) - f(2)}{9 - 2} \]
This expression represents the ratio of the total change in the output \(\Delta f = f(9) - f(2)\) to the total change in the input \(\Delta x = 9 - 2\).
- Identifying the Correct Option and Conclusion
Comparing the derived expression \(\frac{f(9) - f(2)}{9 - 2}\) with the provided options:
A\(f(9) - f(2)\)
B\(\frac{f(9) - f(2)}{9 - 2}\)
C\(\frac{f(9 - 2)}{9 - 2}\)
The derived expression matches option (b). Therefore, the correct expression to determine the average rate of change is \(\frac{f(9) - f(2)}{9 - 2}\).
Examples
The concept of average rate of change is vital in many real-world applications. For instance, if you are tracking the distance \(d(t)\) a car travels over time \(t\), the average rate of change of distance over a specific time interval \([t_1, t_2]\) is simply the average velocity of the car during that trip, calculated as \(\frac{d(t_2) - d(t_1)}{t_2 - t_1}\). This helps engineers and commuters understand performance and plan travel times effectively.